| L(s) = 1 | − 9-s − 12·17-s − 25-s + 16·31-s + 12·41-s − 16·47-s − 14·49-s + 16·71-s + 12·73-s + 16·79-s + 81-s − 20·89-s + 4·97-s − 12·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 2.91·17-s − 1/5·25-s + 2.87·31-s + 1.87·41-s − 2.33·47-s − 2·49-s + 1.89·71-s + 1.40·73-s + 1.80·79-s + 1/9·81-s − 2.11·89-s + 0.406·97-s − 1.12·113-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.455509530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455509530\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.365546814270864509614671861373, −8.340897020749289322554902039409, −8.158708139118437292602012492774, −7.72294749082642464180345614998, −6.93559654804436494363567977821, −6.84720282134134985491516513359, −6.40245765285982593864208682561, −6.29606943066810058409371400870, −5.83266514253386816781244329603, −5.13881258486332263582284633475, −4.74763745368210474400001104672, −4.66519859272480830258016077660, −4.12823505527514364498289153209, −3.75663285447403205674677296201, −2.97490382747463930501203688463, −2.85132397678617966105249341550, −2.10283724795209616844370072338, −2.00331012984413380569161078289, −1.05485074668424738362404367463, −0.38429250094777768150614499986,
0.38429250094777768150614499986, 1.05485074668424738362404367463, 2.00331012984413380569161078289, 2.10283724795209616844370072338, 2.85132397678617966105249341550, 2.97490382747463930501203688463, 3.75663285447403205674677296201, 4.12823505527514364498289153209, 4.66519859272480830258016077660, 4.74763745368210474400001104672, 5.13881258486332263582284633475, 5.83266514253386816781244329603, 6.29606943066810058409371400870, 6.40245765285982593864208682561, 6.84720282134134985491516513359, 6.93559654804436494363567977821, 7.72294749082642464180345614998, 8.158708139118437292602012492774, 8.340897020749289322554902039409, 8.365546814270864509614671861373
